///*
// * Licensed to the Apache Software Foundation (ASF) under one or more
// * contributor license agreements.  See the NOTICE file distributed with
// * this work for additional information regarding copyright ownership.
// * The ASF licenses this file to You under the Apache License, Version 2.0
// * (the "License"); you may not use this file except in compliance with
// * the License.  You may obtain a copy of the License at
// *
// *      http://www.apache.org/licenses/LICENSE-2.0
// *
// * Unless required by applicable law or agreed to in writing, software
// * distributed under the License is distributed on an "AS IS" BASIS,
// * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// * See the License for the specific language governing permissions and
// * limitations under the License.
// */
//
//package org.apache.commons.math4.legacy.linear;
//
//import org.apache.commons.math4.core.jdkmath.JdkMath;
//
///**
// * Calculates the rectangular Cholesky decomposition of a matrix.
// * <p>The rectangular Cholesky decomposition of a real symmetric positive
// * semidefinite matrix A consists of a rectangular matrix B with the same
// * number of rows such that: A is almost equal to BB<sup>T</sup>, depending
// * on a user-defined tolerance. In a sense, this is the square root of A.</p>
// * <p>The difference with respect to the regular {@link CholeskyDecomposition}
// * is that rows/columns may be permuted (hence the rectangular shape instead
// * of the traditional triangular shape) and there is a threshold to ignore
// * small diagonal elements. This is used for example to generate {@link
// * org.apache.commons.math4.legacy.random.CorrelatedVectorFactory correlated
// * random n-dimensions vectors} in a p-dimension subspace (p &lt; n).
// * In other words, it allows generating random vectors from a covariance
// * matrix that is only positive semidefinite, and not positive definite.</p>
// * <p>Rectangular Cholesky decomposition is <em>not</em> suited for solving
// * linear systems, so it does not provide any {@link DecompositionSolver
// * decomposition solver}.</p>
// *
// * @see <a href="http://mathworld.wolfram.com/CholeskyDecomposition.html">MathWorld</a>
// * @see <a href="http://en.wikipedia.org/wiki/Cholesky_decomposition">Wikipedia</a>
// * @since 2.0 (changed to concrete class in 3.0)
// */
//public class RectangularCholeskyDecomposition {
//
//    /** Permutated Cholesky root of the symmetric positive semidefinite matrix. */
//    private final RealMatrix root;
//
//    /** Rank of the symmetric positive semidefinite matrix. */
//    private int rank;
//
//    /**
//     * Decompose a symmetric positive semidefinite matrix.
//     * <p>
//     * <b>Note:</b> this constructor follows the linpack method to detect dependent
//     * columns by proceeding with the Cholesky algorithm until a nonpositive diagonal
//     * element is encountered.
//     *
//     * @see <a href="http://eprints.ma.man.ac.uk/1193/01/covered/MIMS_ep2008_56.pdf">
//     * Analysis of the Cholesky Decomposition of a Semi-definite Matrix</a>
//     *
//     * @param matrix Symmetric positive semidefinite matrix.
//     * @exception NonPositiveDefiniteMatrixException if the matrix is not
//     * positive semidefinite.
//     * @since 3.1
//     */
//    public RectangularCholeskyDecomposition(RealMatrix matrix)
//        throws NonPositiveDefiniteMatrixException {
//        this(matrix, 0);
//    }
//
//    /**
//     * Decompose a symmetric positive semidefinite matrix.
//     *
//     * @param matrix Symmetric positive semidefinite matrix.
//     * @param small Diagonal elements threshold under which columns are
//     * considered to be dependent on previous ones and are discarded.
//     * @exception NonPositiveDefiniteMatrixException if the matrix is not
//     * positive semidefinite.
//     */
//    public RectangularCholeskyDecomposition(RealMatrix matrix, double small)
//        throws NonPositiveDefiniteMatrixException {
//
//        final int order = matrix.getRowDimension();
//        final double[][] c = matrix.getData();
//        final double[][] b = new double[order][order];
//
//        int[] index = new int[order];
//        for (int i = 0; i < order; ++i) {
//            index[i] = i;
//        }
//
//        int r = 0;
//        for (boolean loop = true; loop;) {
//
//            // find maximal diagonal element
//            int swapR = r;
//            for (int i = r + 1; i < order; ++i) {
//                int ii  = index[i];
//                int isr = index[swapR];
//                if (c[ii][ii] > c[isr][isr]) {
//                    swapR = i;
//                }
//            }
//
//
//            // swap elements
//            if (swapR != r) {
//                final int tmpIndex    = index[r];
//                index[r]              = index[swapR];
//                index[swapR]          = tmpIndex;
//                final double[] tmpRow = b[r];
//                b[r]                  = b[swapR];
//                b[swapR]              = tmpRow;
//            }
//
//            // check diagonal element
//            int ir = index[r];
//            if (c[ir][ir] <= small) {
//
//                if (r == 0) {
//                    throw new NonPositiveDefiniteMatrixException(c[ir][ir], ir, small);
//                }
//
//                // check remaining diagonal elements
//                for (int i = r; i < order; ++i) {
//                    if (c[index[i]][index[i]] < -small) {
//                        // there is at least one sufficiently negative diagonal element,
//                        // the symmetric positive semidefinite matrix is wrong
//                        throw new NonPositiveDefiniteMatrixException(c[index[i]][index[i]], i, small);
//                    }
//                }
//
//                // all remaining diagonal elements are close to zero, we consider we have
//                // found the rank of the symmetric positive semidefinite matrix
//                loop = false;
//            } else {
//
//                // transform the matrix
//                final double sqrt = JdkMath.sqrt(c[ir][ir]);
//                b[r][r] = sqrt;
//                final double inverse  = 1 / sqrt;
//                final double inverse2 = 1 / c[ir][ir];
//                for (int i = r + 1; i < order; ++i) {
//                    final int ii = index[i];
//                    final double e = inverse * c[ii][ir];
//                    b[i][r] = e;
//                    c[ii][ii] -= c[ii][ir] * c[ii][ir] * inverse2;
//                    for (int j = r + 1; j < i; ++j) {
//                        final int ij = index[j];
//                        final double f = c[ii][ij] - e * b[j][r];
//                        c[ii][ij] = f;
//                        c[ij][ii] = f;
//                    }
//                }
//
//                // prepare next iteration
//                loop = ++r < order;
//            }
//        }
//
//        // build the root matrix
//        rank = r;
//        root = MatrixUtils.createRealMatrix(order, r);
//        for (int i = 0; i < order; ++i) {
//            for (int j = 0; j < r; ++j) {
//                root.setEntry(index[i], j, b[i][j]);
//            }
//        }
//    }
//
//    /** Get the root of the covariance matrix.
//     * The root is the rectangular matrix <code>B</code> such that
//     * the covariance matrix is equal to <code>B.B<sup>T</sup></code>
//     * @return root of the square matrix
//     * @see #getRank()
//     */
//    public RealMatrix getRootMatrix() {
//        return root;
//    }
//
//    /** Get the rank of the symmetric positive semidefinite matrix.
//     * The r is the number of independent rows in the symmetric positive semidefinite
//     * matrix, it is also the number of columns of the rectangular
//     * matrix of the decomposition.
//     * @return r of the square matrix.
//     * @see #getRootMatrix()
//     */
//    public int getRank() {
//        return rank;
//    }
//}
